Homodyne detection on GKP qubits alone does not give us access to the non-Pauli measurements used in the original proposal [1]}.
It is therefore necessary for us to inject physical magic states \(\vert m \rangle = \left|0\right\rangle _{\rm gkp} + e^{i\frac{\pi }{4}} \left|1\right\rangle _{\rm gkp}\)  created by the GKP factories into the RHG lattice.
This process – illustrated in Fig:injection – requires some classical post-selection (which can be performed in parallel to ensure a high probability of success) and a subsequent \(d\)  rounds of error correction [2]}.
In our case, the error correction is performed after the patch merging step of the multi-qubit Pauli measurement, as described in the previous paragraph.
